Comments

The symmetric difference operation is associative, i.e. $A \Delta (B \Delta C) = (A \Delta B) \Delta C$, and intersection is distributive over it, i.e. $A \cap (B \Delta C) = (A \cap B) \Delta (A \cap C)$. Thus, $\Delta$ and $\cap$ define a ring structure on the power set $\mathcal{P}(X)$ of a set $X$ (the set of subsets of $X$), in contrast to union and intersection. This ring is the same as the ring of $\mathbb{Z}/2\mathbb{Z}$-valued functions on $X$ (with pointwise multiplication and addition). Cf. also Boolean algebra and Boolean ring for the symmetric difference operation in an arbitrary Boolean algebra.